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March 13, 2005

Derived categories for dummies, Part I

Posted by Urs Schreiber

For my own good here I’ll try to review a couple of key ideas. Corrections are welcome.

First of all: what is a derived category anyway?

I realize that there is lots of mathematical literature on this question which I am totally ignorant of. The following is what I extracted from Aspinwall’s discussion.

The short answer is: The derived category D(𝒞) of any abelian category 𝒞 is the category of complexes in 𝒞.

An abelian category is a category equipped with the necessary structure so as to make it possible to have sequences of morphisms in that category which form a complex. So in an abelian category 𝒞 we can have sequences

(1)ℰ•=⋯→dn−1 ℰn→dnℰn+1 →dn+1 ℰn+2 →dn+2 ⋯

with all the ℰn being objects and all the dn, n∈ℕ being morphisms in 𝒞 and such that the composition of any two subsequent morphisms ‘is zero’ in a sense.

Furthermore, we can speak of the n-th cohomology group

(2)ℋn(ℰ•)

of such a complex. This is essentially the usual definition of cohomology suitably formulated so as to apply to general abelian categories.

These complexes ℰ• are the objects of the derived category D(𝒞). There is an obvious notion of morphism between such complexes, and these will indeed be morphisms in D(𝒞). But D(𝒞) has more morphisms than just the obvious ones, and that takes a small amount of preparation to define:

There is an obvious and natural notion of a morphism between two complexes known as a chain map

(3)f:ℰ•→ℱ•

which consists of morphisms

(4)fn:ℰn→ℱn

such that every diagram in sight commutes.

There are 2-morphisms between chain maps known as chain homotopies. Two chain maps related by a chain homotopy are ‘essentially the same’. All this is precisely as in the ‘ordinary’ (non-category theory) context. (Maybe if I find more time later I’ll spell out more details here.)

From any chain map f:ℰ•→ℱ• between the complexes ℰ• and ℱ• one gets an induced map

(5)f*n:ℋn(ℰ•)→ℋn(ℱ•)

between the cohomologies of these complexes.

Note that f* contains less information than the f it comes from. Due to that, f* may be invertible (be an isomorphism) while f itself is not. In this case f is called a quasi isomorphism. In the derived category D(𝒞) we declare these quasi-isomorphisms to be true isomorphisms, i.e. we declare that the morphisms in D(𝒞) are

The crucial point of derived categories is that D(𝒞) knows how to subtract or annihilate objects of 𝒞. 𝒞, being abelian, is an additive category, but need not have a notion of subtraction.

In order to get that, one needs to make use of the translation functor

(6)T:D(𝒞)→D(𝒞)

which shifts all the indices n of the complexes by one n→n+1 .

Given two complexes, we can take their direct sum

(7)ℰ•⊕ℱ•=⋯→ℰn⊕ℱn→dnℰ⊕dnℱℰn+1 ⊕ℱn+1 →⋯.

But given a morphism

(8)f:ℰ•→T(ℱ•)

we can replace the map

(9)dnℰ⊕dnℱ=[dnℰ0 0 dnℱ]

with the map

(10)Cone(f)=[dnℰ0 fdnℱ].

Let me denote the result of this operation as

(11)ominusf.

This corresponds to subtraction because, as one can show

(12)ℰ•ominus1 ℰ•

is isomorphic to the trivial 0-complex (which is the 0-object in D(𝒞)).

This has a very nice physical interpretation:

D-branes carry gauge fields which live in fiber bundles which can be described by their sheaves of sections. If 𝒞 is the category of (‘coherent’) sheaves then the objects in D(𝒞) can be shown to describe general configurations of D-branes, up to some technical fine print. If ℰ• describes some D-brane then T(ℰ•) describes its anti-D-brane and

(13)ℰ•ominusfℰ•∼0

describes the mutual annihilation of the two, whith the off-diagonal map f descibing the tachyonic strings which stretch from ℰ• to T(ℰ•) and which communicate the annihilation process.

So much for now, typing this stuff always takes longer than expected, even when already expected to take somewhat longer. More later, if time permits.

Here is a question that I would like to know the answer to:

Consider a quiver Q and its category 𝒞=Rep(Q) of representations. 𝒞 is a 2-algebra. Is the derived category D(𝒞)=D(Rep(Q)) still a 2-algebra in an appropriately weakened sense? In particular, does the tensor product in D(𝒞) distribute (up to isomorphism) over the subtraction operation ominus1 ?

I’d expect this to be true, but I have not yet figured out the details. I think this might be important because given a dimensional deconstruction of a (2,0 ) theory by means of a ‘large’ quiver Q a natural guess for the vector 2-bundles expected to play a role in the (2,0 ) theory are modules of some 2-algebra naturally associated to Q, and the obvious guess is to use D(Rep(Q)) – instead of Rep(Q) as I guessed previously.

Posted at March 13, 2005 6:43 PM UTC

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MathML

For some reason, at least on my system, whenever I had a \mathcal{C} in the above formulas what appears on my screen is just a question mark. I didn’t have this problem before. Sorry about that. If anyone knows how to fix it please let me know.

Monoidal Derived Categories

USENET is sooo slow. Over on sci.math.research I posted a silly question on monoidal derived categories. Of course it was Aaron who replied and set me straight. But since my followup to that has still not appeared and since it fits well under the headline ‘derived categories for dummies’, I’ll post it here:

Thanks. I should have known that. As described for instance in Mac Lane’s
‘Homology’ p. 163.

So the unit e in D(C) is the complex with the unit of C in the 0th position
and zeros everywhere else.

There is sort of a ‘negative unit’, too, namely T(e), where T is the
translation functor. Multiplying with T(e) is the same as acting with T. Of
course its square is unity only up to even translation.

Is anything known about (weak) multiplicative inverses in D(C)??

The reason why I am asking is this:

Let Q be the quiver with a single non-identity edge. Then Rep(Q) is like the categorification of the natural numbers. It should be straightforward to throw in additive inverses to get what I called RepI(Q)recently
which is sort of a categorification of the integers. Both Rep(Q) and
RepI(Q) sort of sit inside D(Rep(Q)), which is much larger. Does the latter maybe contain multiplicative inverses?

Here is another question:

Given D-branes E and F described by complexes E• and F• of coherent sheaves. Is there any natural physical picture associated to forming E•⊗F•?

Like there is a nice physical picture to taking the “sum” of E and F using
cones, which corresponds to putting the stacks of branes on top of each
other and turning on string states going between them.

Re: Monoidal Derived Categories

Remember, otimes is not an operation in the derived category. Instead, its left derived functor is. This is usually denoted by an otimes with an L on top of it. You have to take projective resolutions to define it. (See Gelfand and Manin or Weibel).

Unfortunately, I’m too doped up on cold medication right now to think about the answers to your questions.

we can find what is called a set ℛ of adapted objects (adapted to F that is) having some properties. The category K+(ℛ) of (bounded from below complexes of) these objects (with quasi-isos inverted) is equivalent to the bounded derived category D+(𝒜).

The point seems to be that this way we can talk about 𝒜 by restricting attention to the (smaller) ℛ.

And because of that we want to apply the functor F not on 𝒜 but really on ℛ, in a sense. In any case the right derived functorRF of F is defined as the composition

(2)RF=D+(𝒜)→K+(ℛ)/(quasi-isos)→FD+(ℬ).

This RF now is exact.

(I am writing this just in order to better remember it. I know that I am not even scartching the surface yet.)

I had a couple of other things to do today, but I managed to go to the library and get lots of books. I took

Hilton & Stammbach: A course in homological algebra

Northcott: Homological algebra

Cartan & Eilenberg: Homological algebra

Weibel: An introduction to homological algebra

and

Mac Lane: Homology .

Since I’ll leave for Vietri tomorrow evening and I want to use the 20 hour train ride for some reading, can you suggest any other books/papers that I should make sure to have a look at?

Re: Monoidal Derived Categories

Gelfand and Manin, _Methods of Homological Algebra_ is the best I’ve run across for derived categories. Weibel is great for derived functors, but the derived category stuff is only a single chapter at the end.

Note that there’s another book by Gelfand and Manin called “Homological Algebra” which is not the book I’m talking about.

Also, with the proviso is that you shouldn’t really trust anything I say in my medication induced stupor, I think that if you have an identity for the tensor product in your Abelian category, the complex with that object concentrated in degree zero should be an identity for the left-derived tensor product in the derived category. Unless I’m missing something, it’s pretty obvious: you take a CE resolution of the object you want to tensor with and tensor away. You get back your resolution which is quasiisomorphic to your original complex, so you’ve got what you started with up to isomorphism.

Re: Derived categories for dummies, Part I

Consider a quiver Q and its category
Rep(Q) of representations. Rep(Q) is
a 2-algebra. Is the derived category
D(Rep(Q)) still a 2-algebra in
an appropriately weakened sense?

I think I know how you’re taking the tensor product of two quiver representations. A quiver representation is a functor

F: Q -> Vect

and if we another one, say

G: Q -> Vect

you can form the composite

Q -> Q x Q -FxG-> Vect x Vect -tensor-> Vect

and get a representation you call F tensor G. Here the first arrow is the “diagonal” functor.

(I’m not gonna try that MathML crap yet - I just want to see if this post works!)

Anyway, assuming this is right, I guess now you’re gonna plunge in and study quiver representations on chain complexes, and then see if the tensor product of these gets along with the model category structure on chain complexes so that we can get a “monoidal model category”.

(This is about 5 times more sophisticated than it needs to be, since people were studying derived categories and chain complexes long before they invented model categories… but heck! Let’s show off!)

Anyway, this has got to work. You define
a quiver representation on a chain complex to be a functor

F: Q -> ChainComplexes

and define the tensor product of such guys just as we did for representations on vector spaces. We’re not using anything about “Vect” or “ChainComplexes” other than it being a monoidal category, so far.

Then the question is whether this tensor product gets along with the model category structure on chain complexes - most importantly, the quasiisomorphisms.

Have you figured this out yet, or should I cheat by looking up some stuff?

I’m sure it’ll work, and I really think you are following the Tao of Mathematics, going with the grain instead of against it - so you’ll be rewarded by the math gods.

Re: Derived categories for dummies, Part I

Have you figured this out yet, or should I cheat by looking up some stuff?

I haven’t figured out the answer to my original question. Instead I have changed the question until it had an anwer! ;-)

More seriously, I came to think that my original idea to get hold of a 2-algebra here was not the most natural one. I proceeded by the following steps:

We are looking for a categorification of the algebra of functions over some space taking values in some field k. Because that’s hard in general, we might start with the toy example of a discrete space S, which is just a finite set of points. A function on S is a morphism S→fk in Set. Hence its categorification should be a morphism in Cat, i.e. a functor. Hence we must lift S to some category. This is naturally done by stringifying it and throwing in some arrows between the points of S. This yields a quiver Q, which I’ll identitfy with its path category Q.

Next we must find a suitable target category. You have argued in HDA2 for Hilb, which is Vect with extra structure. BD&R have shown that this goes in the right direction, mathematically, and several ideas from string theory, too, support this as a good idea.

So let me consider a functor Q→Vect as the first approximation to the categorification of the algebra of functions in kS. These form the functor category (Vectk)Q which is a nice 2-algebra.

However, when we decategorify this 2-algebra by taking isomorphism classses of objects we see that we get ℕ-valued functions on S. This is not quite what we wanted. Maybe we don’t need to require to really get k-valued functions, but we would like to divide and subtract somehow.

As I have argued, taking the derived category D(VectQ) might be a good idea. When done right it introduces a notion of subtraction, but still not one of division.
Hence we should not be content with D(VectQ)
as our candidate categorification of kS.

I will, however, have good use for D(VectQ) further below, but not as a categorification of a function algebra but as a candidate category of 2-section of a vector 2-bundle over Q!

This comes about as follows:

The state of affairs becomes maybe less murky once we use an equivalent reformulation of VectQ. Namely this category is ‘the same’ (insert the proper adjective here) as the category kQ−Mod of (left,say) modules of the path algebrakQ of the quiver Q.

But once we are talking about modules, there is an obvious notion of potentially invertible product that suggests itself, namely the tensor product ⊗kQ over kQ. Of course for that to work we need to switch from left modules to bimodues.

For this reason I am proposing that we should replace kQ−Mod by kQ−Bimod, the category of bimodules over the quiver path algebra kQ.

I believe using ⊗kQ as the monoidal product this becomes a weak monoidal category. Some of the bimodules in here are ‘tilting elements’ meaning that they are weakly invertible. I believe there should hence be a subcategory of kQ−Bimod with these tilting elements as objects, which is in fact a weak 2-group. But I haven’t really checked the technical fine print. The decategorification of this subcategory however is well known as the Picard group of kQ.

So concerning our categorification program the step from VectQ≃kQ−Mod to kQ−Bimod is much like the step from natural numbers to positive rational numbers, it seems.

Assuming this to be a good idea, we still need to throw in additive inverses. So let’s pull the derived category trick at this point and go from kQ−Bimod to its derived category D(kQ−Bimod).

The details of how to really ‘subtract’ in a triangulated derived category are still being worked out by Bridgeland and others, apparently, but I shall be content with the rough notion of subtraction that we have here. (BTW, is D(kQ−Bimod) triangulated when D(kQ−Mod) is?)

The product ⊗kQ now becomes the derived product ⊗L and with respect to this D(kQ−Bimod) is still monoidal. The weakly invertible objects are now of course called titling complexes. The isomorphism classes of these form what is called the derived Picard group of kQ.

There is a series of very nice papers, talks and lecture notes on this by Amnon Yekutieli.

So that’s currently my proposal:

A ‘good’ candidate for the categorification of the algebra kS of functions on a discrete space S is the derived category D(kQ−Bimod) of bimodules of the path algebra of a quiver whose vertex set is S.

I also have some ideas on how to check that this makes good sense:

If D(kQ−Bimod) is a good ‘2-algebra’, then, in the spirit of 2-NCG/2-SQM, we should find that states of a categorified quantum mechanics are 2-sections of ‘line 2-bundles’ with respect to this 2-algebra, where a ‘line 2-bundle’ here is a ‘1-dimensional’ module of D(kQ−Bimod).

There is an obvious candidate for what such a section should look like ‘locally’, it should just be an object in D(kQ−Mod), since that’s what D(kQ−Bimod) naturally acts on via ⊗L.

In other words, a line 2-bundle over Q would be something that locally looks like D(kQ−Mod).

I have described this idea of how to define vector 2-bundles in more detail in the section ‘Vector 2-bundles’ of these unfinished notes.

The fun thing is, that this turns out to be essentially right! Objects of D(kQ−Mod) are indeed known to describe states in 2-quantum mechanics, namely states of strings attached to D-branes.

And the ‘gauge group’ acting on these line 2-bundles, under which our 2-SQM should be physically invariant, is nothing but the derived Picard (2-)group which, again, is indeed known to describe certain duality operations on string configurations.

The string theory side suggests that we might want to go one more step further and replace D(kQ−Bimod) somehow with derived categories of left/right modules with different path algebras acting on both sides. This is what describes (generalized) ‘Seiberg dualities’ aka tilting equivalences aka mutations in D-brane physics.

And also it is essentially known how everything I wrote down here for representations of quivers translates into categories of coherent sheaves over these Calabi-Yau.

On the other hand, the category-theoretic side here might suggest something about string theory: If the line 2-bundle picture that I have drawn here is to be taken seriuously, then it would follow that a state of D-branes with strings between them is only ‘locally’ in some sense an element of D(kQ−Bimod). Globally several such elements might have to be glued together, since these would be local 2-section, not 2-functions.

I was very lucky to get lots of feedback on this idea from Aaron. He, too, does entertain the idea that there should be some sort of twisting going on for string fields (related to elliptic cohomology) but doesn’t see why the line 2-bundle that I have sketched here is what should capture this.

Possibly he is quite right, and I will have to further think about this stuff.

I am however intrigued by how closely a 2-connection on a line 2-bundle looks like the enriched elliptic objects by Stolz and Teichner.

Namely, if we restrict attention to complexes concentrated in degree 0, then a 2-conneciton would essentially assign elements of the weak Picard 2-group to bigons, i.e. bimodules to edges and bimodule morphisms to surfaces. This is quite like an enriched elliptic object.

One difference is that in the picture sketched above we would naïvely use the ordinary tensor product for horizontal composition, while Stolz and Teichner instead use Connes fusion. As far as I understand, however, Connes fusion is like the ordinary tensor product but ‘corrected by a twist’, somehow, such to ensure that all modules in the game are at the same time Hilbert spaces.

I learned that the fibered category QCoh which assigns to each open set (or scheme) U the category QCoh(U) of quasi-coherent sheaves over U (section 3.2.1) is in fact a stack (theorem 4.23).

I had known about stacks in the context of gerbes, which are stacks in groupoids, but was unaware of the simple fact that we may want to study much more general stacks in ‘higher gauge theory’. A stack in groupoids is essentially the same as a certain class of principal 2-bundles. I’d assume that a more general stack would correspond to some associated 2-bundle.

Since we are dealing with coherent sheaves here this perspective might provide another way to identify a 2-bundle structure in D-brane physics.

Namely what if we generalized the above statement to the derived context? Like what if we had a fibered category that assigned the derived category of (quasi?-)coherent sheaves to each open set?

Zoran Skoda kindly informed me that there is a concept of derived fibered categories (do I recall that correctly?) which should apply here.

It would be cool if the the fibered category in derived coherent sheaves were something like a stack. That should go a long way towards identifying a 2-bundle structure in D-branes physics.

I have tried to find literature on this, but without real success so far.

Update: I am being kindly informed that more information can be found here:

Grothendieck topology

For those following this (anyone? :-), let me briefly mention the idea behind Grothendieck topology:

In low-brow terms a sheaf is something that assigns to each open set of some manifold a certain set, such that when you make your open set smaller there is a prescibed way how to make that set smaller in a ‘globally consistent way’.

The archetypical example is the sheaf of sections of some fiber bundle. To every open set of the base space of some bundle it assigs the set of sections of the bundle restricted to that open set.

What is important about this is that from the sheaf of sections of a bundle the entire bundle can be reconstructed, so that we can alternatively forget about bundles and formulate everything in terms of sheaves.

In more high-brow language we can reformulate the definition of sheaves as follows:

Given any manifold X, we get a category of open sets O(X) in that manifold whose objects are all open sets and which has an arrow A→B precisely if A is a subset of B. Alternatively we can turn all these arrows around to get the opposite category Oop(X).

Using this terminology a sheaf is nothing but a functor

(1)Oop(X)→Set

with extra properties (called, at least once everything is categorified, the descent data) which ensures that everything can be glued together globally.

So in this formulation a sheaf is something you can define on a category of the form O(X).

As always in mathematics, when we have some construction we try to get away with all the unnecessary ingredients and extract the bare essence. Hence we want to know which properties of the categories of open sets are really necessary so have something like a sheaf. The answer is called Grothendieck topology (def. 2.24 in Vistoli’s review). A category is said to have a Grothendieck topology when its objects and morphisms sort of behave as if they were open sets and inclusions of open sets. But they can be something quite more abstract.

So its really a rather simple idea. To hide the simplicity of this idea and to scare people a little a category with Grothendieck topology is called a site. A site is hence something on which you can have a sheaf. (See section 2.3.3 of Vistoli’s review.)

Of course for physicists an entertaining riddle is to think about what exotic string theory setup could possibly give rise to D-branes described by coherent sheaves that are sheaves over a site not of the form O(X). Maybe D-branes on fuzzy spaces or something?

Everybody bored by this account should try to blow his mind with the beautiful

Read the post Kamnitzer on Categorifying Tangle InvariantsWeblog: The n-Category CaféExcerpt: A brief summary of a talk by Joel Kamnitzer at the workshop on Categorification and Geometrisation from Representation Theory.Tracked: April 14, 2009 1:41 PM